Problem 66 - Thomas Calculus

Question

Which series in Exercises \(53-76\) converge, and which diverge? Give reasons for your answers. If a series converges, find its sum. $$\sum_{n=1}^{\infty} \frac{n^{n}}{n !}$$

Step-by-Step Solution

Answer

The series diverges by the Ratio Test.

Step 1: Define the Series

The series in question is \( \sum_{n=1}^{\infty} \frac{n^n}{n!} \). We need to determine if this series converges or diverges.

Step 2: Apply the Ratio Test

To determine convergence, we use the Ratio Test. The Ratio Test states that a series \( \sum a_n \) converges absolutely if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \).For our series, \( a_n = \frac{n^n}{n!} \). We need to find:\[\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{(n+1)^{n+1}}{(n+1)!} \cdot \frac{n!}{n^n} \right| \]

Step 3: Simplify the Limit Expression

Let's simplify the expression:1. \( \frac{(n+1)^{n+1}}{n^n} = \frac{(n+1)^{n} \cdot (n+1)}{n^n} = \left(\frac{n+1}{n}\right)^n \cdot (n+1) \)2. \( \frac{n!}{(n+1)!} = \frac{1}{n+1} \)3. Combining these gives \[\lim_{n \to \infty} \left| \frac{(n+1)^{n+1}}{(n+1)!} \cdot \frac{n!}{n^n} \right| = \lim_{n \to \infty} \left(\frac{n+1}{n}\right)^n = e\].

Step 4: Analyze the Limit Result

The limit found in Step 3 is \( e \), which is approximately 2.718...Since \( e > 1 \), according to the Ratio Test, the series \( \sum_{n=1}^{\infty} \frac{n^n}{n!} \) diverges.

Key Concepts

Understanding Series Convergence

In mathematics, determining whether an infinite series converges or diverges is a fundamental concept. Convergence refers to the property of a series to approach a specific value as more and more terms are added. On the other hand, divergence means that the series does not settle towards any particular value. For an infinite series, written as \( \sum_{n=1}^{\infty} a_n \), we explore these properties to understand if the series is summing up to a finite number or not.

• If the series converges, the sum \( S \) can be found where \( S = \lim_{n \to \infty} S_n \) with \( S_n \) being the partial sum of the first \( n \) terms.
• A diverging series means \( \lim_{n \to \infty} S_n \) does not exist or becomes infinite.

Tests like the Ratio Test, discussed later, are crucial in determining these properties of series.

Demystifying Factorial Functions

The factorial function, denoted as \( n! \) (read as 'n factorial'), is a key mathematical operation, especially in series and combinatorics. It is defined as the product of an integer and all the positive integers below it:

• \( n! = n \times (n-1) \times (n-2) \times ... \times 1 \)
• For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• By convention, \( 0! = 1 \).

Factorials grow at an extraordinary rate, often overpowering exponential and polynomial functions in terms of size. This growth behavior of factorial functions is pivotal when analyzing series and other mathematical expressions that contain factorials. The comparison of \( n! \) to other parts of a series can significantly influence whether a series converges or diverges.

Exploring the Limit Comparison Test

The Limit Comparison Test is a handy tool in determining the behavior of series. It helps compare a given series with another series whose convergence behavior is known:

• We compare the series \( \sum a_n \) to \( \sum b_n \), where \( b_n \) is a benchmark series.
• If \( \lim_{n \to \infty} \frac{a_n}{b_n} = c \), where \( 0 < c < \infty \), both series will either converge or diverge together.
• This method is particularly useful when direct application of tests, like the Ratio Test or Integral Test, is complex.

Understanding and using the Limit Comparison Test can greatly simplify the analysis of series convergence, providing a shortcut by leveraging known results of similar series.

Diving into Infinite Series

An infinite series is a sum of infinitely many terms like \( \sum_{n=1}^{\infty} a_n \). These are not just theoretical constructs; they play crucial roles in mathematical analysis, physics, and engineering.

• Infinite series can represent complex functions and model real-world phenomena.
• Some well-known infinite series include geometric series and harmonic series.
• The behavior of these series is often studied to understand their convergence properties.

Working with infinite series involves using various tests to interrogate their sum behaviors. Convergence tests, such as the Ratio Test and Comparison Tests, are strategic tools to unravel the mysteries of infinite series. Mastery of these principles allows one to predict whether these series tend towards a sum total or escape into infinity.